Ridge regression signal processing for position-fix navigation systems

ABSTRACT

A position estimator for determining the position and velocity of a moving platform in cooperation with radio navigation aids is described incorporating an unbiased estimator, such as a least means square estimator, a biased estimator for determining the angle of inner section of the lines of position from the radio naviation aids for determining the liklihood of geometric dilution of precision (GDOP) and a switch for selecting the estimate of position and velocity from said biased estimator at first times and the unbiased estimator at second times. The invention overcomes the problem of accuracy degredation associated with a nearly collinear measurement geometry which causes the variance of the position estimates to be higly inflated.

BACKGROUND OF THE INVENTION

1. Field of the Invention:

This invention relates to position-fix navigation systems and moreparticularly to calculating target position and velocity from sensorinformation with reduced mathematical variance at times the sensors arenearly collinear with the target, a condition known as the geometricdilution of precision (GDOP).

2. Description of the Prior Art:

All position-location navigation schemes can be classified as deadreckoning position-fix or combinations of both. A position, once known,can be carried forward indefinitely by keeping continuous account of thevelocity of the vehicle, ship or aircraft. This process is called deadreckoning.

A position-fix navigation system, in contrast to dead reckoning,determines the position of a vehicle, ship or aircraft without referenceto any former position. The simplest position fix stems from theobservation of a recognizable landmark. More frequently, however, radioaids to navigation provide position information where the physicallocations of the radio aids, the ground station transmitters, are known.At present, most aircraft in the world use distance measuring equipment(DME) and very high frequency omnirange (VOR) which provide position-fixinformation and wherein the physical location of the DME or VOR areknown. Other position-fix systems are LORAN-C and Omega as well as therecently developed position-fix systems such as the Global PositioningSystem (GPS) and the Relative Navigation Component of the Joint TacticalInformation Distribution System (JTIDS).

The aircraft position-fixing is traditionally accomplished bydetermining the intersection of two or more lines of position (LOP) withrespect to some known reference system. These LOP's represent theintersection of three-dimensional surfaces of position with the earth'ssurface which in many practical cases is approximated locally by aplane. Thus, on the earth's surface, the LOP or a range measurement is acircle while an angle (bearing) measurement is a line. A position-fix isthe intersection of two LOP's which require at least two measurements.Since no measurement can be made without error, the LOP's will have anerror associated therewith and thus the intersection of two LOP's willgenerate a position error.

The position error may be substantially increased at times the LOP's arenearly collinear (i.e. not orthogonal). This position error mechanism isknown as the geometric dilution of precision (GDOP). One example wherethe LOP's are nearly collinear is when an aircraft is between twolandmarks along a line between the landmarks or when the aircraft isoutside both landmarks along a line between both landmarks. Anotherexample of GDOP is when the aircraft distance from two landmarks isgreat compared to the distance between the two landmarks. Position errordue to GDOP begins to occur when the angle γ between the intersection oftwo LOP's is greater that 150° or less than 30°.

Virtually all aircraft position-fix algorithms use some form of unbiasedLeast Mean Squared Estimation (LMS). The position of the aircraft isexpressed as a plurality of linear equations which are then solved forthe unknown values of position in coordinates (X and Y) and velocityhaving components V_(x) and V_(y). The linear equations may be expressedin the form

    Y=Xβ+e                                                (1)

where Y is the nxl observation vector, X is an nxp prediction matrix ande is the nxl error vector with covariance matrix, W. The statisticalproblem is how to best guess the component values of the pxl regressorvector β. As is well known, one method is the ordinary LMS solutionwhich assumes the covariance matrix W as given in Equation (4). It isgiven in Equation 2,

    β.sub.OLS =(X.sup.T X).sup.-1 X Y                     (2)

which follows from the so called "normal" equations, shown in Equation3,

    X.sup.T Xβ.sub.OLS =X.sup.T Y                         (3)

Equation (3) assumes that

    W=σ.sup.2 I                                          (4)

where σ² is the variance (i.e. σ is the standard deviation) and I is thenxn identity matrix. When ##EQU1## the LMS solution β_(GEN) is calledGeneralized least square where

    β.sub.GEN =(X.sup.T W.sup.-1 X).sup.-1 X.sup.T W.sup.-1 Y (4.2)

A most critical performance measure is the variance of the estimate; itis given by

    VAR(β.sub.OLS)=(X.sup.T X).sup.-1 σ.sup.2       ( 4.3)

for the OLS estimate.

When GDOP exists, the sensor errors i.e. receiver noise, quantizationnoise, propagation effects, etc. can be "blown up" or "blown down" whenthe sensor measurements are referenced to the navigation coordinates todetermine an aircraft position fix. This inflation of the positionestimate errors (GDOP) arises mathematically from the X^(T) X term inEquation 4.3. It may have values which are very small or values whichare equal to zero in the diagonal terms of the matrix. These unusuallysmall or zero values "blow up" to be very large values when the inverseof X^(T) X (i.e. (X^(T) X)³¹ 1 is calculated as indicated in Equation4.3. The meaning of Equation 4.3 is that on the average the very largevalues in the matrix (X^(T) X)⁻¹ inlate the values of the ordinary LMSsolution β when Equation 2 is solved.

Present navigation systems on aircraft accept the position estimates,some of which are highly inflated.

Mathematically, a plurality of linear equations may be solved whereinthe relationship X^(T) X is used where X^(T) is the transpose of thematrix X and where X^(T) X is a matrix having eigenvalues λ. When therelationship X^(T) X moves from a unit matrix to one where highmulticollinearity exists, variance inflation will rise as the smallesteigenvalue λs approaches zero. A mathematical technique whichcounteracts the effects of multicollinearity was disclosed in apublication by J. Riley wherein the diagonal terms of the X^(T) X matrixwere limited to certain minimum values. This technique avoided numericaldifficulties when inverting a square matrix. The publication by J. Rileyis entitled "Solving Systems of Linear Equations--With a PositiveDefinite, Symetric But Possibly Ill-conditioned Matrix," MathematicTables and Other Aids to Compute., Vol. 9, 1955. In a furtherdevelopment, a paper was published by A. Hoerl and R. Kennard entitled"Optimum Solution of Many Variable Equations" Chemical EngineeringProgress, Vol. 55, No. 11, November 1959. Two additional papers werepublished by A. Hoerl and R. Kennard entitled "Ridge Regression and BiasEstimation for Nonorthogonal Problems", Technometrics, Vol. 12, No. 1,February 1970 and "Ridge Regression: Applications to NonorthogonalProblems", Technometrics, Vol. 12, No. 1, February 1970. The authorsdescribed a technique termed "Ridge Regression" which counteracts theeffects of multicollinearity. In Ridge Regression, the diagonalcomponents of a matrix are limited to a predetermined value by adding asmall term K to each diagonal component. That is X^(T) X is transformedinto X^(T) X+KI where I is the unit matrix. The idea in Ridge Regressionis to find an estimator whose variance decreases as [X^(T) X+KI]⁻¹ ;which means that the variance inflation is limited to 1/K even when theeigenvalues of X^(T) X→0. Equation 5 shows the expression for the RidgeRegression coefficients β_(R).

    β.sub.R =[X.sup.T X+KI].sup.-1 X.sup.T Y.             (5)

In the literature, K is known as the Ridge parameter.

SUMMARY OF THE INVENTION

An apparatus and method is described for determining the position andvelocity of a moving platform comprising a radio navigation aid havingat least three portions, two of which are distant from the platform andhaving a known position for providing to the platform a plurality ofsamples indicative of the position of the platform at respective times,a position estimator for receiving the plurality of samples forgenerating an estimate of position and velocity, both with an unbiasedestimator and with a biased estimator and selecting the estimate ofposition and velocity from the biased estimator at times the geometry ofthe moving platform is substantially collinear with the two distantportions of the radio navigation aid.

It is an object of this invention to provide a position estimator whichutilizes a biased estimator based upon Ridge Regression at times thegeometry of the navigation landmarks are nearly collinear.

It is a further object of this invention to provide a position estimatorutilizing a least mean square estimator at first times and a RidgeRegression estimator at second times when the geometric dilution ofprecision (GDOP) is present.

It is a further object of this invention to provide a position estimatorsuitable for functioning in the guidance loop in the navigation systemof an aircraft, for example, wherein position samples are provided by aposition-fix navigation system such as a distance measuring equipment(DME), global positioning system (GPS), very high frequency omnirange(VOR) etc. and wherein a Ridge Regression algorithm is used at timesGDOP is present.

Brief Description of the Drawings

FIG. 1 is a block diagram of a typical navigation system for anaircraft.

FIG. 2 is a block diagram of one embodiment of the invention.

FIG. 3 is a diagram showing the intersection of two lines of position ina position-fix navigation system.

FIG. 4 is a diagram showing the intersection of two lines of positionintersecting at a small angle in a position-fix navigation system.

FIG. 5 is a diagram showing position samples taken along a flight path.

FIG. 6 is a diagram showing two position samples along a flight path.

FIG. 7 is a graph of the Bias, Standard Deviation and (MSE)^(1/2) versusthe Ridge Variable K.

FIG. 8 is a graph of the Ridge Regression coefficient versus the RidgeVariable K.

FIG. 9 is a graph of the error ellipsoid for initial position estimates.

FIGS. 10 and 11 are scatter plots of 1000 β_(OLS) 16-sample estimates.

FIGS. 12 and 13 are scatter plots of 1000 β_(R) 16-sample estimates.

FIG. 14 is a scatter plot of β_(OLS) estimates for Y-Y equal to aconstant.

FIG. 15 is a diagram showing the intersection of two lines of position.

FIG. 16A is a scatter diagram of 1000 sets, each set including 16 pairsof range measurements.

FIG. 16B is a rotated and expanded portion of FIG. 16A.

FIG. 17 is a scatter diagram of initial aircraft position where eachdata point represents a 30 second time average.

FIG. 18 is a scatter diagram of the 1000 Ridge estimates showing thebias error at the center of the estimates with respect to the truevalue.

FIG. 19 is a portion of FIG. 18 expanded.

FIG. 20 is a diagram showing the error ellipsoid of the data plotted inFIG. 16A.

FIG. 21 is a diagram showing cononical rotation of the error eppilsoidcoordinates.

FIG. 22 is a diagram showing angles of lines of position.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, a typical navigation system 10 of an aircraft 26 isshown. A commanded position for a desired flight path position iscoupled over lead 12 to an input of adder 14. The position of aircraft26 is sensed by radio navigation aids such as distance measuringequipment (DME) or very high frequency omnirange (VOR). Radio navaid 16has an input from aircraft 26 over lead 18 indicative of aircraftposition and an output over lead 20 which may provide a plurality ofsamples indicative of the position of aircraft 26 at respective times.Lead 20 is coupled to an input of airborne position estimator 21 whichfunctions to receive the plurality of samples indicative of the positionof aircraft 26 at respective times and to generate an estimate ofposition and velocity of the aircraft. Airborne position estimator 21typically is a least mean square (LMS) estimator. The output of airborneposition estimator is coupled over lead 22 to a second input of adder14. Adder 14 functions to combine mathematically the commanded positionon lead 12 with the estimated position on lead 22 to provide a steeringcommand or error signal over lead 23 to an input of controller 24.Controller 24 may be, for example, an autopilot or human pilot whichcouples a correction signal to the aircraft. The correction signal ismaintained near its null position by changing the direction of theaircraft's velocity vector by controlling the attitude of the aircraft.Inducing a roll angle command causes the aircraft to turn left or rightfrom its flight path. Similarly, a change in its pitch angle causes theaircraft to fly above or below its intended flight path. The correctionsignal is coupled over lead 25 to aircraft 26. The attitude of aircraft26 is sensed by airframe attitude sensors 27 which may have an inputover lead 28 from aircraft 26. The output of airframe attitude sensors27 are coupled over lead 29 to a second input of controller 24.

In FIG. 1, the control loop shown by leads 28 and 29 provide feedback tocontroller 24 and is commonly known as the flight control system loop.The outer loop shown by leads 18, 20 and 22 provide feedback informationto adder 14 as to the position of the aircraft and is commonly known asthe guidance control loop. The output of airborne position estimator 21on lead 22 is dependent upon the performance of radio navaid 16 andairborne position estimator 21. As determined by the desired flight pathor track over the earth's surface and the position of other distantportions of radio navaid 16 which may have transmitter locations on theearth's surface, the performance of radio navaid 16 may be degraded dueto collinearity of the distant portions of radio navaid 16 and aircraft26. For example, radio navaid 16 may provide two lines of position (LOP)from two transmitters having respective locations on the earth'ssurface. The LOPs may intersect at either a large angle greater than150° or at a small angle less that 30° which results in nearly collinearmeasurement geometry termed the general dilution of precision (GDOP).The degraded performance from radio navaid 16 on line 20 is processed byairborne position estimator 21 to provide degraded position informationon lead 22 at times collinear measurement geometry or GDOP is present.The degraded position information on lead 22 results in degradedsteering commands on lead 23 to controller 24 causing controller 24 tovary the velocity vector of aircraft 26 away from the desired flightpath at times GDOP is present.

Referring to FIG. 2, a block diagram of position estimator 31 is shownsuitable for use in place of airborne position estimator 21 shown inFIG. 1. Radio navaid 16 which may have at least three physical portions,two of which are distant from the aircraft and which have a knownposition such as a respective transmitter location on the earth'ssurface. A third position of navaid 16 is positioned on the aircraft orother platform such as a helicopter, ship, land vehicle, etc. The outputof radio navaid 16 provides a plurality of samples indicative of theposition of the aircraft at respective times which may be, for example,spaced apart in time by an intersample period ΔT_(S) The plurality ofsamples may all be taken within a time interval ΔT_(M). Lead 20 iscoupled to an input of geometric collinear estimator 34 which has storedtherein the known positions of the radio navaid 16 portions which aredistant from aircraft 26. Geometric collinear estimator 34 functions tocalculate the angle between intersecting lines of position from at leasttwo distant portions of radio navaid 16 and aircraft 26. At time theangle γ between the intersecting lines of position at aircraft 26 isless than or equal to, for example, 30° or greater than or equal to150°, the output of the geometric collinear estimator 34 on lead 31 willgo high causing switch 36 to couple lead 38 to lead 22. At other timesswitch 36 will couple lead 40 to lead 22.

The output of radio navaid 16 is also coupled over lead 20 to an inputof biased estimator 42 and to an input of unbiased estimator 43.Unbiased estimator 43 may be, for example, a least mean square estimatorto provide position and velocity estimates as a function of time overlead 40 to an input of switch 36 and to an input of K parameter 44. Kparameter 44 functions to provide a bias value k over lead 46 to aninput of biased estimator 42. An optimum K value may be derived, forexample, from equation 6. ##EQU2## In equation 6, P is the number ofvariables and S² is the sample standard deviation.

Biased estimator 42 may be, for example, a Ridge estimator having a biasvalue K as shown in equation 7. As noted previously, K is called theRidge parameter in the literature.

    β.sub.R =(H.sup.T H+K.sub.OP I).sup.-1 H.sup.T δR (7)

A biased estimator 42 or Ridge estimator can have a smaller mean squareerror than an unbiased estimator 43, because the biased estimator 42relaxes the unbiased condition. It allows a small bias error in order toachieve a large variance reduction. The result is a smaller mean squareerror than that achieved by conventional least mean square (LMS)estimation.

FIG. 3 is a diagram showing the intersection of two lines of position(LOPs) 48 and 49 in a position-fix navigation system having distancemeasuring equipment 50 and 51 spaced apart from one another by adistance shown by arrow 52. In FIG. 3, the LOPs intersect at an angle γwhich is optimum for precise measurement when γ is equal to or near 90°.The angle γ is shown by arrow 53. LOP 49 traces on an arc 54 having aspread in possible values of measurement shown by lines 55 and 56. LOP48 traces on an arc 57 having a spread in possibles values ofmeasurement shown by arcs 58 and 59. The point of intersection of arcs54 and 57 show the position of aircraft 26 and the area enclosed by line55, 58, 56 and 59 shown as area 60 shows the area of possible values ofposition measurement of aircraft 26.

FIG. 4 is a diagram showing the intersection of two lines of position 48and 49 at a small angle γ in a position-fix navigation system. In FIG. 4like references are used for lines of position and apparatus as used inFIG. 3. FIG. 4 shows an X and Y horizontal coordinate system having anorigin 0,0. DME 50 is located at (-150000;0). DME 51 is located at150,000'. The lines of position 48 and 49 are shown intersecting ataircraft 26 at an angle of 6.1° as shown by arrow 53. Radio navaid 16would provide n samples in time interval ΔT_(M) while aircraft 26travels a short distance shown by arrow 61. Aircraft 26 may be, forexample, at an initial horizontal position of X=-400,000'and Y=50,000'.Aircraft 26 may be flying on a straight course at constant velocity Vover the time interval ΔT_(M).

FIG. 4 also shows aircraft 26 at a later time at a position X,Y of(-141,061; 50,000). Lines of position of 48' and 49' to aircraft 26 fromDME 50 and 51, respectively form an angle of 90° at their intersectionat aircraft 26. An angle of 90° provides the most favorable measurementconditions for measuring aircraft position. Subsequent in time asaircraft 26 flies along to position X, Y of (400,000; 50,000) the linesof position 48" and 49" will again intersect at an angle of 6.1°. As theaircraft continues to fly in the direction past 400,000', the angle γwill continue to decrease.

The intersection of the lines of position 48 and 49 shown in FIG. 4 maybe found by solving equations 8 and 9 for X₀ and Y₀.

    R.sub.1.sup.2 =(X-X.sub.C1).sup.2 +(Y-Y.sub.C1).sup.2      (8)

    R.sub.2.sup.2 =(X-X.sub.C2).sup.2 +(Y-Y.sub.C2).sup.2      (9)

The solution for X₀ and X₀ is shown in equations 10 and 11 where R₁ isthe distance to DME 50 and R₂ is the distance to DME 51 and X_(C1) isthe position of DME 50. ##EQU3## By utilizing equations 10 and 11, theinitial position of aircraft 26 may be determined and geometriccollinear estimator 34 shown in FIG. 2 may calculate the angle formed bythe intersection of the lines of position 48 and 49.

FIG. 5 is a diagram showing the position of aircraft 26 along a desiredflight path 63. DME 50 and DME 51 enable radio navaid 16 to generateposition samples 47, and 64-72 shown along flight path 63 having anintersample time ΔT_(S) shown by arrow 73. The number of samples thatmay be processed at one time may be, for example, sixteen samplesassociated with DME 50 and sixteen samples associated with DME 51. Thesesamples are taken over a time interval ΔT_(M) with the assumption thatthe partial derivatives of R₁ and R₂ with respect to X do not changeappreciatively. For processing the block of data, 32 samples, it isfurther assumed that the aircraft is flying a straight course atconstant velocity over the measurement interval ΔT_(M). Additionalblocks of data later in time may be obtained by radio navaid 16 foraircraft 26 at positions 72, 74, 75 along flight path 63.

Referring to FIG. 6, a diagram showing two position samples 76 and 47 isshown along with lines of position 77 and 48 from DME 50. The flightpath segment 78 extends between position sample 76 and 47. Flight pathsegment 78 is more generally described as R. By using the well knownTaylor series linearization of non-linear geometry, R may be expressedas shown in equation 12. In equation 12, ₁ is shown in FIG. 6 by arrow79 which is measured from a horizontal line in a counterclockwisedirection to flight path segment 78, R. ##EQU4## A linear model ofaircraft position and movement is obtained by differentiating equations8 and 9 to obtain equation 13. ##EQU5## In Equation 13 ##EQU6## is thenoise source ##EQU7## is the known deterministic part.

The aircraft position at the ith measurement is given by Equation 14 and15.

    X.sub.i =X.sub.1 +iV.sub.x ΔT.sub.s ; i=0, 1, 2. . . n-1 (14)

    Yi=Y.sub.1 +iV.sub.Y ΔT.sub.s                        (15)

After n measurements the initial estimate position is given by Equations16 and 17.

    X.sub.1 =X.sub.o +δX                                 (16)

    Y.sub.1 =Y.sub.o +δY                                 (17)

The estimated position at t_(o) +T_(M) is given by Equations 18 and 19.

    X.sub.n =X.sub.1 +(n-1)V.sub.x ΔT.sub.s              (18)

    Y.sub.n =Y.sub.1 +(n-1)V.sub.Y ΔT.sub.s              (19)

The estimated aircraft speed is given by Equation 20. ##EQU8## For theith measurement let δR_(i1) =R_(i1) -R₀₁ and δR_(i2) =R_(i2) -R₀₂. Aftern pairs of measurements the linear model equation is given by Equation21. ##EQU9## where [e₁₁, e₁₂ . . . e_(n1), e_(n2) ]=e^(T) is the errorvector. The error vector has a covariance given by Equations 22 and 23.##EQU10## Equation 21 may be rewritten as Equation 24.

    δR=Hβ+e                                         (24)

In Equation 24 δR is the 2n×1 measurement vector, H is the 2n×4predicator matrix, β is the 4×1 regression coefficient vector. Thenotation has changed from the usual statistical symbols as used inEquation (1) to avoid confusion with the notation used in the navigationgeometry. Using FIGS. 5 and 6 and Equation 12, the Taylor coefficientsare given in Equations 25-28. ##EQU11## The average of 30 pairs of rangemeasurements (R₀₁, R₀₂), as shown n FIG. 5, using DME 50 and DME 51 maybe made to determine the initial point X_(o) Y_(o) at t=t₁ usingEquations 10 and 11.

The unbiased estimator 42 shown in FIG. 2 may use a least mean square(LMS) solution to equation 21 which was rewritten as equation 24. TheLMS solution to equation 21 or 24 is given by equation 29.

    β.sub.OLS =(H.sup.T W.sup.-1 H).sup.-1 H.sup.T W.sup.-1 δR (29)

In equation 29 the conditions shown in equation 30 are assumed.##EQU12## In Equation 29, H^(T) H will now be given in Equation 31 usingEquations 21 and 25-28 and assuming, ##EQU13## From Equations 21 and25-28, H^(T) R is given in Equation 32. ##EQU14##

In equations 29-31 and by letting the covariance W equal I the identitymatrix, equation 29 may be rewritten as equation 33.

    β.sub.OLS =(H.sup.T H).sup.-1 H.sup.T δR        (33)

wherein the calculation of equation 33 may be facilitated by usingequations 31 and 32. In equation 33 β_(OLS) is the ordinary least squaresolution for the vector as shown in equation 34. The vector given inequation 34 is provided on line 40 from unbiased estimator 43 shown inFIG. 2.

    β.sub.OLS =[δX, δY, V.sub.X, V.sub.Y ].sup.T(34)

Biased estimator 42 may use a Ridge Regression solution to solveEquations 21 or 24 as shown in Equation 35.

    β.sub.R =(H.sup.T H.sup.-1 H+KI).sup.-1 H.sup.T W.sup.-1 δR (35)

By letting the covariance W equal I where I is the identify matrix,equation 35 may be written as shown in Equation 36.

    β.sub.R =(H.sup.T H+KI).sup.-1 H.sup.T δR       (36)

In equation 36, K is supplied by K parameter 44 over line 46 which isshown in FIG. 2. I is the identity matrix which is a square matrix withzeros except along the diagonal which are ones. Equation 36 may now besolved using Equations 31 and 32. The solution of Equation 36 is avector shown in Equation 35.

    β.sub.R =[δX, δY, V.sub.x, V.sub.y ].sup.T (37)

K parameter 44 functions to supply a value K to biased estimator 42which is used when biased estimator 42 solves equation 36. K parameter44 may be, for example, the Ridge parameter K. An automatic method forchoosing K_(OP) is given in Equation 38. ##EQU15##

In Equation 38 P is the number of variables in the linear model (P=4)and S₂ is the sample standard deviation based upon (N-P) samples (i.e.28). S² may be determined using equation 39. ##EQU16##

Alternatively, K parameter 44 may select a K value which is passed on tobiased estimator 42 to calculate the corresponding Ridge estimatesβ_(R). The K parameter 44 may submit incrementally increasing K valuesto biased estimator 42 which will calculate the corresponding Ridgeestimates β_(R). The optimum K value would be the one at which the Ridgeestimates stabilize with small changes is K as shown in FIG. 7.

The results are plotted in FIG. 8; note that as expected K=0.1 is nearthe minimum MSE. The data for FIG. 8 is given in Table IA.

                  TABLE 1A                                                        ______________________________________                                                δx                                                                      1.0E = 003*                                                                            δy  V.sub.x V.sub.y                                    ______________________________________                                        K = 0.00  -0.7868    0.9910    3.7272                                                                              0.3047                                             0.1787     0.8507    1.6541                                                                              0.6055                                             0.3610     0.8246    1.1636                                                                              0.6754                                             0.4205     0.8164    0.9357                                                                              0.7070                                             0.4410     0.8138    0.8003                                                                              0.7252                                   K = 0.05  0.4453     0.8137    0.7090                                                                              0.7369                                             0.4422     0.8146    0.6422                                                                              0.7451                                             0.4356     0.8160    0.5907                                                                              0.7510                                             0.4272     0.8177    0.5495                                                                              0.7555                                             0.4181     0.8195    0.5156                                                                              0.7589                                   K = 0.10  0.4087     0.8213    0.4871                                                                              0.7615                                             0.3995     0.8230    0.4627                                                                              0.7636                                   ______________________________________                                    

The contour ellipsoid for the position coordinates can be determinedusing H_(p) which is a partition of the matrix H as given by equation21. The details are given in FIG. 9. The subspace for the velocityestimates can also be analyzed using FIG. 9. Table IB summarizes theresults wherein the Ridge Regression technique yielded useable positionestimates where γ=6.1°, whereas the LMS estimates given in Table IB arenot operationally useable.

                                      TABLE IB                                    __________________________________________________________________________    γ = 6.1°                                                                            LMS ESTIMATE                                                                            RIDGE ESTIMATE                                   TRUE VALUE         k = 0     k = 0.1                                          __________________________________________________________________________    δ.sub.X                                                                      1000 FT       -786 FT   418 FT                                           δ.sub.Y                                                                      1000 FT       991 FT    819 FT                                           V.sub.X                                                                            727 FT/SEC    3727 FT/SEC                                                                             515 FT/SEC                                       V.sub.Y                                                                            420 FT/SEC    304 FT/SEC                                                                              758 FT/SEC                                       NOISE                                                                              STANDARD DEVIATION                                                                          SAMPLE                                                     MODEL                                                                              (SD) = 600    SD = 550.4 FT                                                                 MEAN = 100.7 FT                                            __________________________________________________________________________

In fact, even though the LMS condition index was moderate (19.8), one ofthe position estimates had the wrong sign.

Table II summarizes the results wherein the Ridge Regression techniqueand the LMS technique yielded useable position estimates where γ=90°.

                                      TABLE II                                    __________________________________________________________________________    γ = 90°                                                                             LMS ESTIMATE                                                                            RIDGE ESTIMATE                                   TRUE VALUE         k = 0     k = 0.1                                          __________________________________________________________________________    δ.sub.X                                                                      1000 FT       1045 FT   1043 FT                                          δ.sub.Y                                                                      1000 FT       717 FT    718 FT                                           V.sub.X                                                                            727 FT/SEC    625 FT/SEC                                                                              616 FT/SEC                                       V.sub.Y                                                                            420 FT/SEC    504 FT/SEC                                                                              493 FT/SEC                                       NOISE                                                                              STANDARD DEVIATION                                                                          SD = 550.4 FT                                                   (SD) = 600    MEAN = 100.7 FT                                            __________________________________________________________________________

A computer simulation was performed in which 1000 sample pairs of sizeN=16 were taken. From these samples 1000 estimates were determined andplotted in FIGS. 10-13. FIGS. 10 and 11 show a plot of the ordinary LMSestimate for δX, δY, V_(x) and V_(y) while FIGS. 12 and 13 show theRidge estimates for K=0.1. FIGS. 10-13 were obtained using equation 39.5where Q is a general point in the estimation space.

    YQ.sup.2 =YP.sup.2 +PQ.sup.2                               (39.5)

The four dimension estimation space has been partitional into an initialposition space shown in FIGS. 10 and 12 and the velocity space shown inFIGS. 11 and 13. These scatter plots are computer simulationrealizations of the concept illustrated in FIG. 14. For thesesimulations the true position of the aircraft was at (-400,000',50,000') The combined calibration and initial position errors were[δX,δY]=[2500, 2200] and the noise standard deviation was 600 feet. Theaircraft was flying from the (-400,000', 50,000') waypoint at 30° with avelocity of 840 feet/second. Thus, the true speeds were V_(x) =727feet/second and V_(y) =420 feet/second. As is evident in the scatterplots of FIGS. 12 and 13, the GDOP has been reduced without introducingunacceptable bias. Equally important they demonstrate that the Ridgeestimates confidence region are well within the LMS confidence regions.The scatter plots of FIGS. 12 and 13 support the major result of RidgeAnalysis, that is by selecting the Ridge parameter K at that point wherethe β_(R) estimates are stable estimates will be produced whosevariances (and MSE) are less than or equal to the LMS estimates. Theabove analysis addressed the estimates of δX, δY, V_(x) and V_(y). Theseestimates are then inserted into equations 14 and 15 to obtain theaircraft's position at ΔT_(M). An analysis of these final positionestimates X_(n), Y_(n) was not performed. The calculations, however, arestraight-forward.

The model described and analyzed above is valid wherever the aircraftvelocity has a constant direction and its speed remains essentiallyconstant over the measurement interval of ΔT_(M). The model can begeneralized to include acceleration maneuvers.

The second case will now be analyzed. When the aircraft position is atposition 47 (-400,000, 50,000) as indicated in FIG. 4, multicollinearity(GDOP) should be clearly present because γ=6.1°. Table III presents thefirst four terms (out of 16) of the H matrix given by Equation 21.

                  TABLE III                                                       ______________________________________                                                  δ.sub.X                                                                        δ.sub.Y                                                                          V.sub.X  V.sub.Y                                    ______________________________________                                        H = PREDICTOR MATRIX                                                          First 4 Pairs of                                                                          0.1959   0.9806   0      0                                        Measurements                                                                              0.0924   0.9957   0      0                                                    0.1959   0.9806   0.0122 0.0613                                   H =         0.0924   0.9957   0.0058 0.0622                                               0.1959   0.9806   0.0245 0.1226                                               0.0924   0.9957   0.0115 0.1245                                               0.1959   0.9806   0.0367 0.1839                                               0.0924   0.9957   0.0173 0.1867                                               0.7508   4.5460   0.3520 2.1309                                               4.5460   31.2492  2.1309 14.6480                                  H.sup.T H = 0.3520   2.1309   0.2273 1.3762                                               2.1309   14.6480  1.3762 9.4602                                   V = EIGENVECTOR                                                                           0.1459   1.0000   -0.5025                                                                              -0.0733                                  V =         1.0000   -0.1459  0.0733 -0.5025                                              0.0733   0.5025   1.0000 0.1459                                               0.5025   -0.0733  -0.1459                                                                              1.0000                                   L = EIGENVALUES                                                                           39.4294  0        0      0                                        L =         0        0.1083   0      0                                                    0        0        0.0059 0                                                    0        0        0      2.1440                                   Eigenvalues of                                                                Unit Scaled X'X                                                                           3.590    0.287    0.114  0.009                                    Condition                                                                     Indices     1.000    3.537    5.614  19.856                                   ______________________________________                                    

Also shown are the values for the H^(T) H matrix, its eigenvectors V andits eigenvalues L. The matrix H^(T) was calculated using equation 31. Asample of 16 pairs of range measurements (R₁ and R₂) were taken todetermine B_(R). The true initial correction is X, Y=1000,1000. (TableIII)

The regression coefficients were calculated for values of K ranging fromK=0 (LMS) to K=0.5. Clearly, the optimum choice for minimum bias, isK=0.1 as shown in FIG. 7. At this value the Ridge traces have juststabilized and the variance improvement with increasing K is stillovercoming the rate of bias increase. This point is evident when the MSEis plotted as a function of K. The bias (β_(R)), √variance (β_(R)) and√MSE (β_(R)) were calculated using equations 40 and 44 where

    VAR[β.sub.R ]=G.sub.K X.sup.T VAR[Y]XG.sub.k.sup.T    (40)

    VAR[β.sub.R ]=σ.sup.2 G.sub.K X.sup.T XG.sub.K.sup.T (41)

    VAR[β.sub.R ]=σ.sup.2 X.sup.T X[X.sup.T X+KI].sup.-2 (42)

    MSE[β.sub.R ]=E[(β.sub.R -β).sup.T (β.sub.R -β)](43)

    MSE[β.sub.R ]=TRACE[σ.sup.2 X.sup.T X[X.sup.T X+KI].sup.-2 ]+K.sup.2 β.sup.T [X.sup.T X+KI].sup.-2 β       (44)

In equations 40 through 44, equations 45 through 47 were used.

    β.sub.R =[δX, δY, V.sub.x, V.sub.y ]      (45)

    β.sup.T =[1000, 1000, 727, 420]                       (46)

    δ=600 feet.                                          (47)

A method and apparatus has been described for determining the positionand velocity of a moving platform comprising a radio navigation aidhaving at least three portions, two of which are distant from theplatform and having respective positions for providing to the platform aplurality of samples indicative of the position of the platform atrespective times, a position estimator for receiving the plurality ofsamples for generating an estimate of position and velocity both with anunbiased estimator and with a biased estimator and selecting theestimate of position and velocity from the biased estimator at times thespatial geometry of the moving platform is substantially collinear withthe two portions of the distant radio navigation aid.

A RIDGE REGRESSION NAVIGATION EXAMPLE

Due to the complexity of the subject, additional information is providedbelow as an aid to understanding the properties of a biased estimator.The discussion below will,

(a) assume the aircraft speed is small so that velocity calculations arenot necessary. This will simplify Equation 21.

(b) review the biased/unbiased estimation procedures using the "positiononly" equations,

(c) describe the procedure for obtaining the initial position estimate,

(d) determine whether GDOP exists; if GDOP is present, use the biasedestimator (i.e., the Ridge Estimator)

(e) in the context of the simple example, perform the detailedcalculations to obtain the results.

The Ridge Regression algorithm is implemented as follows:

(1) Linearize Locus of Position Equations (Eq. 8 and 9).

(2) Determine initial aircraft position using DMS estimation on 30second of data.

(3) Estimate aircraft position using Ridge Regression in one secondestimates using previous estimates as the new approximate position ofthe aircraft.

The equations of position are given in (8 and 9). Their linearizedmodels in terms of a.c range (R₁, R₂) and its displacement (δX, δY) froman initial position (X_(o), Y_(o)) are: ##EQU17##

R₀₁ and R_(o2) are the initial approximate ranges to the aircraft. Fromthe navigation geometry shown in FIG. 15, the partial derivatives in(48) and (49) are: ##EQU18##

Let

    δR.sub.1 =R.sub.1 -R.sub.01                          (50.1)

where R₁ and R₂ are measured values

    δR.sub.2 =R.sub.2 -R.sub.02                          (50.2)

Then (48) and (49) become ##EQU19##

Equation (51) constitutes one pair of measurements. The examplenavigation system repeats these measurements 16 times in one second,therefore, ##EQU20## δR is a 32×1 vector, H_(p) is a 32×2 matrix and δβis a 2×1 vector.

As noted earlier, Equation (51) only incorporates the positioninformation, so that the essential Ridge Regression ideas become morevisible. The complete expression including the velocity terms is givenin (21). The aircraft position is determined by first guessing theaircraft's position and then using the unbiased or biased algorithmdepending upon the existence GDOP (see FIG. 2) (e.g. Ridge Regression)to calculate the correction δβ to the initial guess to determine theaircraft's true position. Since the aircraft's velocity is not beingdetermined at this time, it is assumed that its speed is small.Therefore,

(A) Guess the aircraft's present position using (X_(o), Y_(o)) where(X_(t), Y_(t)) is its true position. (X_(o), Y_(o)) is approximated bytaking 30 apirs of range measurements and calculating the sample meansR_(o1) and R_(o2).

(B) Insert R_(o1) and R_(o2) into Equations 53a and 53b to obtain(X_(o), Y_(o)). The equations relating to R_(o1), R_(o2), X_(cl),Y_(cl), X_(c2), Y_(c2) to (X_(o), Y_(o)) ##EQU21##

(C) R_(o1) and R_(o2) are next subtracted from the 16 pairs of measuredvalues to obtain δR as indicated in (52).

(D) Since H_(p) is given by the coordinate geometry and δR has beendetermined from (C) above, Equation 52 can be solved to obtain δβ. Thecomputed values of ##EQU22## are then added to (X_(o), Y_(o)) to obtainthe true position estimate of the aircraft (X_(t), Y_(t)). This positionestimate is then used as the "next" guess of the aircraft's position,and "new" correction terms are then estimated to calculate the "nest"position of the aircraft as it flies along its intended course.

A detailed example will now be given which demonstrates the improvedperformance of Ridge Regression over LMS. Assume the aircraft is at theposition given in FIG. 15, aircraft coordinates are (-262,448,50,000).Assumes further that the pilot assumes his position at (-270,148,51,500)which is offset by (-7700,1500) from the aircraft's actual position. Theproblem then is to estimate the (-7700,1500) offset and subtract it from(-270,148,15,500) to obtain the correct position of (-262,448,50,000).These calculations are repeated using the next sequence of measurements.

The error model is Equation (51) wherein 16 pairs of measurements havebeen made in one second. Using FIG. 15, the data matrix H_(p) has thevalue, ##EQU23##

It is at this point that the check for GDOP is made. As will be notedlater, this check embodies calculating the determinant of (H_(p) ^(T)H_(p))⁻¹. That is, GDOP is present if, ##EQU24## and are proportional tothe major and minor axis of the error ellipsoid as shown in FIG. 9 (note|H_(p) ^(T) H_(p) | is the determinant of the matrix [H_(p) ^(T) H_(p) ]

It will be shown later that ##EQU25## When there is no GDO, |H_(p) ^(T)H_(p) |⁻¹ =1 because when there is GDOP, and. This means that 1/λ₂ >>1,and consequently |H_(p) ^(T) H_(p) |⁻¹ >>1. As shown in FIG. 1a, the noGDOP condition results in a circular error ellipse. When GDOP ispresent, 1/λ₂ >>1 and the error ellipsoid becomes elongated. UsingEquation (54), the system model is ##EQU26##

Equation 57 will be used to determine the error ellipsoid (scatter plot)of 1000 position estimate trails. The scatter plot shape also revealswhether GDOP is present. Assuming the initial position of the aircraftto be (-7700,1500), 1000 sets of 16 pairs of measurements were made. Theresults are shown in FIG. 16A. FIG. 16B is a rotated version of FIG. 16Awith expanded scales so that analysis results can be clearly indicated.The ratio of major ellipse axis to the minor ellipse axis is 115:1. Theno-GDOP condition is 1:1. The results are shown in FIG. 16B. As shown,the errors due to GDOP are very large (±80,000. ±15,000). That is, thecorrection estimate of the true offset (-7700,1500) can be as much as 10times the magnitude of the true correction. It is the intent of theRidge Estimates to reduce this wide variation of the estimates. Recallthat the scatter plot (FIG. 16A) is what one can expect for the modelassumed.

Before presenting the Ridge Regression alogrithm it is necessary todetermine a method for estimating the initial aircraft position. This isan important consideration because the Ridge estimator is a biasedestimator. Clearly, one should use an unbiased estimator such as the LMSestimator. But, as shown in FIG. 16, the initial estimate could, forexample, be (-87,700, 16,500) which because of the GDOP is much toogross an estimate, particularly for a biased estimator. Below it will beshown that the estimation bias is proportional to the error of theinitial guess. An analysis below will illustrate this point.

The idea is to average the one second estimates over a 30 second timeinterval, each individual estimate may have 16 pairs of measurements. A30 second time interval is operationally permissible when the aircraftis beginning its mission. Moreover, the estimate will improve as theflight continues. Namely, the measurement process is convergent. Theresults of a 30 second average are shown in FIG. 17 using the LMS dataof FIG. 16A. The range of the 33 sets of measurements have been reducedfrom (± -80,000, 15000 to ±7700, ±1500). The reduction is consistentwith the sample variance reduction factor which is proportional to 1/30.The standard deviation reduction is √1/30 ≃1/5₅

One of the worst case values will be used as the initial guess of theaircraft's position, namely, (-7700, 1500). Since severe GDOP has beendetected, the switch in FIG. 1 selects the biased estimator (Ridgeestimator). The Ridge estimate is given by

    δβ.sub.R =[H.sub.p.sup.T H.sub.p +kI].sup.-1 H.sub.p δR (58)

where for this example k=0.05.

The analysis will proceed inserting the same data as used in the FIG. 16experiment. The results are given in FIG. 18. As indicated, the variancehas shrunk considerably. See FIG. 19 which is a "blow up" of the 1000Ridge measurements. The price for this variance shrinkage was a biaserror of (7343, -1402). That is, the average estimate of the offset(-7700, 1500) was (357,92), but the individual measurements were greatlyimproved because the total error is the sum of the bias component and afluctuating component (variance). The bias error will, however, decreasewith each one second measurement. After about 30 seconds, the error willbe reduced to less than 1770 feet. Again, recall that the initial offsetwas a worst case guess.

Another method for reducing the bias of the initial value of the Ridgeestimate is the Jack Knife, which is a boot strap technique. (BradleyEforon, 1981, "The Jack Knife Estimate of Variance", Annals ofStatistics, No. 9, Pgs. 586-596)

In the next section, the nature of the Ridge Regression varianceshrinkage and its bias error will be analyzed using transforming thecoordinate system to the principle axes of the error ellipsoid shown inFIG. 20.

The transformation which rotates the axes from the (X, Y) system to X',Y') system is given by ##EQU27## where U^(T) U=U⁻¹ U=I and φ=θ+0.5° asshown in the figure. Thus, U causes a coordinate rotation of -degreesand U^(T) causes a rotation of -φ°. See FIG. 21. The transformation Uyields the following new vectors and matrices referenced to the newcoordinate system (X¹, Y¹).

    H.sub.C =H.sub.p U                                         (60.1)

    β.sub.C =U.sup.T β                               (60.2)

where H_(C) and β_(C) are the data matrix and the estimation vectorreferenced in the canonical coordinate system. The important quantity isH_(p) ^(T) H_(p) because is expresses the magnitude of GDOP effects. Ofequal important is the fact that the U transformation rotates thecoordinate system such that (X¹, Y¹) lie along the principal axis of theerror illipsoid. For that unique condition, the off-diagonal elements ofH_(p) ^(T) H_(p) are zero and the diagonal elements are called theeigenvalues and the columns of U are called the eigenvectors. Benefitsaccrued from working in the canonical coordinate system (or theeigensystem) is that (1) the effects of GDOP become obvious and (2) thecalculations required to obtain the estimates can be performed almost byinspection. The H_(p) ^(T) H_(p) matrix in the canonical coordinatessystem is ##EQU28## where λ₁ and λ₂ are the eigenvalues given by##EQU29## The inverse of H_(C) ^(T) H is a measure of the GDOP.##EQU30## Thus, if one of the eigenvalues is small (e.g. λ₂ <<¹) thedeterminant of [H_(C) ^(T) H_(C) ] is

    det [H.sub.C.sup.T H.sub.C ]=|H.sub.C.sup.T H.sub.C |=1/λ,λ.sub.2 >>1

That is [H_(C) ^(T) H_(C) ] blows up or it amplifies the variance as isillustrated in FIGS. 9 and 20. Thus, if λ₂ e.g. is small, then thecanonical coordinate ##EQU31## becomes large, generating a highlyelongated ellipsoid. For example, when γ=1° as given in Equation (62).##EQU32## and b/a=115.5. As noted earlier, GDOP generates a highlyelongated ellipse whose b axis is 115.5 times larger than the a axis.

The properties of the LMS estimates will not be viewed in terms of thecanonical coordinate system. Recall that the model for the ithmeasurement pair is

    δR.sub.i =Hβ.sub.T +e.sub.i where E [e.sub.i ]=0 and VAR [e.sub.i ]=σ.sup.2 I                                (65)

and β_(T) is the true offset of the assumed aircraft position from itsactual position. The LMS estimate is ##EQU33## for 16 pairs ofmeasurements. The error in the estimate is ##EQU34## Insert (65) into(67) yielding ##EQU35## The average error of (16) is

    E[β.sub.LMS.sup.-β T]=0 because E[e]=0           (70)

Therefore, the LMS estimate is unbiased. Its variance is ##EQU36## Thus,the variance of β_(LMS) is proportional to GDOP. Its effects areexpressed by the error ellipsoid given in FIGS. 9 and 20.

Clearly the effects of GDOP can be limited if the eigenvalues (λ₁ and λ₂are limited in how small a value they can assume. The Ridge regressionnotion is to add a small quantity k to the diagonal terms of H_(p) ^(T)H_(p). Let ##EQU37## Transform H_(R) ^(T) H_(R) to the canonicalcoordinate system. ##EQU38##

Take the inverse of (H_(R) ^(C))^(T) H_(R) ^(C) ##EQU39## for k=0.05.

The one sigma magnitude of the error ellipsoid axes now become ##EQU40##Thus, the elongation of the error ellipsoid has been reduced by a factorof 24,480/1339 =18.3:1. This reduction is reflected in FIG. 18. Note,however, that a bias has been generated; it will be discussed shortly. Ageometric interpretation can be given to variance reduction caused bythe addition of k to the diagonal terms of H_(p) ^(T) H_(p). Equation 74can be rewritten as ##EQU41## and because 0.95=cos γ, then γ=18.2°. Thusas shown in FIG. 22, the addition of k in H_(p) ^(T) H_(p) has theeffect of limiting how small γ can be decreased, or it is equivalent toexpanding from 1° to 18.2°. The penalty is, of course, a bias error,which simply arises due to the unbalancing of H_(p) ^(T) H_(p) caused bythe insertion of k. This modification is termed Ridge Regression. Forthe Ridge Estimator let the estimation error be,

Ridge True ##EQU42## where the model is

    δR=H.sub.p β.sub.True +e

Make the transformation from the (X, Y) coordinates to the (X¹, Y¹)coordinates using H_(c) =H_(p) U. Equation 77 becomes ##EQU43## where##EQU44## The bias error in the canonical coordinates is where E[e]=0

Insert (59) into (80) using k=0.05, β_(True) =[-7700, 1500] ##EQU45##

Thus, ##EQU46## in the canonical coordinate system.

In the original (X, Y) system the bias error is ##EQU47## The value in(82) is also the one obtained in the computer simulation plotted inFIGS. 18 and 19.

The variance of the Ridge Regression Estimator is given by (83) in thecanonical coordinate system where β_(T) =β_(True). ##EQU48## where theeigenvalues λ₁ and λ₂ are given Equation 62.

The standard deviation is ##EQU49## or VAR(δX)=414 and VAR(δY)=146

The calculated standard deviations given by (85) are consistent with the1000 computer derived samples plotted in FIG. 19.

The invention claimed is:
 1. Apparatus for determining the position andvelocity of a moving platform comprising:a radio navigation aid havingat least three portions, two of which are distant from said platform andhaving a known position for providing to said platform a plurality ofsamples indicative of the position of said platform at respective times,a position estimator for receiving said plurality of samples forgenerating an estimate of position and velocity both with an unbiasedestimator wherein the mean square error of the estimate is the varianceand with a biased estimator and selecting said estimate of position andvelocity from said biased estimator at times said geometry of saidmoving platform is substantially co-linear with said two distantportions of said radio navigation aid.
 2. The apparatus of claim 1wherein said biased estimator is a Ridge estimator.
 3. The apparatus ofclaim 1 wherein said position estimator includes a switch coupled to theoutputs of said unbiased estimator and said estimator biased forcoupling one of said outputs to an output terminal of said positionestimator in response to a control signal.
 4. The apparatus of claim 3wherein said position estimator further includes a geometric collinearestimator for determining the presence of collinearity having an outputcoupled to the control input of said switch.
 5. The apparatus of claim 4wherein said geometric collinear estimator estimates the angle betweentwo lines of position and for providing an output signal to the controlinput of said switch at times said angle is less than a firstpredetermined value and at times said angle is greater than a secondpredetermined value.
 6. The apparatus of claim 1 wherein said positionestimator includes a means for calculating the Ridge parameter K havinga input coupled to the output of an unbiased estimator and wherein theoutput of said means for calculating is coupled to an input of said biasestimator.
 7. The apparatus of claim 1 further including an addercoupled to the output of said position estimator having a second inputadapted for coupling to position commands, the output of said addercoupled to the input of an autopilot, said autopilot adapted forcoupling to said moving platform to direct said moving platform along adisired flight path.
 8. Apparatus for determining the position andvelocity of a moving platform in cooperation with at least two distantradio navigation aids spaced apart in distance by a predetermined amountcomprising:means for receiving a plurality of samples from said radionavigation aids indicative of the position of said platform atrespective times, means for determining the lines of position to eachnavigation aid and the angle of intersection of said lines of positionat said platform, means for generating an estimate of position andvelocity from said plurality of samples with an unbiased estimatorwherein the mean square error of the estimate is the variance, means forgenerating an estimate of position and velocity with a biased estimator,and means for selecting said estimate of position and velocity from saidbiased estimator at times said angle of intersection of said lines ofposition are less than a first predetermined value or greater than asecond predetermined value and for selecting said estimate of positionand velocity from said unbiased estimator at other times.
 9. Theapparatus of claim 8 further including means for calculating a Ridgeparameter K as a function of the output of said unbiased estimator, saidRidge parameter K coupled to an input of said biased estimator.
 10. Theapparatus of claim 8 wherein said biased estimator is a Ridge estimator.11. A method for determining the position and velocity of a movingplatform in cooperation with at least two distant radio navigation aidsspaced apart in distance by a predetermined amount comprising the stepsof:receiving a plurality of samples from said radio navigation aidsindicative of the position of said platform at respective times,determining the lines of position to each navigation aid and the angleof intersection of said lines of position at said platform, generatingan estimate of position and velocity from said plurality of samples withan unbiased estimator wherein the mean square error of the estimate isthe variance, generating an estimate of position and velocity with abiased estimator, and selecting said estimate of position and velocityfrom siad biased estimator at times said angle of intersection of saidlines of position are less than a first predetermined value or greaterthan a second predetermined value and for selecting said estimate ofposition and velocity from said unbiased estimator at other times. 12.The method of claim 11 including calculating a Ridge parameter K as afunction of the output of said unbiased estimator, said Ridge parameterK coupled to an input of said biased estimator.
 13. Apparatus fordetermining the position and velocity of a moving platform in aposition-fix navigation system having known first and second transmitterlocations at times the geometry of said transmitter locations withrespect to the estimated position of the receiver on a moving platformis nearly collinear comprising:first means for receiving and storing aplurality of n samples indicative of range each spaced apart by anintersample time interval ΔT_(S) and occurring within time intervalΔT_(M) from each of said first and second transmitters having a knowndistance therebetween, second means for determining the initial positionX_(O), Y_(O) of said platform from said plurality of n samples from eachof said first and second transmitters where ##EQU50## where R₀₁ is thedistance to the first transmitter at time T₁, R₀₂ is the distance to thesecond transmitter at time T₂,and X_(C1) is the position of said firsttransmitter, third means for determining ##EQU51## +10 where ##EQU52##fourth means for determining H^(T) _(H) where ##EQU53## +10 where A=cos²74 ₁ +cos² θ₂ B=cosθ₁ sinθ₁ +cosθ₂ sinθ₂ C=sin² θ₁ +sin² θ₂ fifth meansfor determining H^(T) H⁻¹, sixth means for determining H^(T) δR##EQU54## seventh means for determining β_(OLS) where β_(OLS) =(H^(T)H)⁻¹ H^(T) δR, eighth means for determining β_(OLS) ^(T), ninth meansfor selecting K_(OP) where ##EQU55## +10where P is the number ofvariables and S² is the sample standard deviation where ##EQU56## +10andtenth means for determining β_(R) where β_(R) =(H^(T) H+K_(OP) I)⁻¹H^(T) δR to provide [δX, δY, V_(x), V_(y) ].